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arxiv: 1907.00789 · v1 · pith:CQXKPID3new · submitted 2019-06-28 · 📡 eess.SP · cs.IT· math.IT

Phase Modulated Communication with Low-Resolution ADCs

Samiru Gayan , Hazer Inaltekin , Rajitha Senanayake , Jamie Evans This is my paper

Pith reviewed 2026-05-25 13:52 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT
keywords low-resolution ADCM-PSKsymbol error probabilityfading channelsquantizationasymptotic analysismaximum likelihood detectionNakagami-m fading
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The pith

With n at least log base 2 of (M plus 1) bits, M-PSK in fading achieves the same symbol error probability decay rate m as infinite-resolution quantization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first derives a universal lower bound on average symbol error probability that holds for any M-ary modulation when the quantizer cannot resolve all signal points. For the special case of M-PSK it constructs the optimal maximum-likelihood detector under equiprobable symbols and circularly symmetric fading, then obtains a closed-form average SEP expression. Analysis and Nakagami-m simulations establish that the high-SNR decay exponent of this SEP equals the fading parameter m precisely when n meets or exceeds log2(M+1), matches the infinite-bit case, and drops to m/2 or zero for smaller n. This matters because it shows how many ADC bits are actually needed to retain full reliability order in fading.

Core claim

A transceiver architecture with n-bit quantization is asymptotically optimum in terms of communication reliability if n is greater than or equal to log2(M+1). That is, the decay exponent for the average SEP is the same and equal to m with infinite-bit and n-bit quantizers for n greater than or equal to log2(M+1). On the other hand, it is only equal to half and 0 for n = log2(M) and n < log2(M), respectively.

What carries the argument

The optimum maximum-likelihood detector for equiprobable M-PSK points under n-bit quantization and circularly symmetric fading, which yields the general average SEP expression.

If this is right

  • When n >= log2(M+1) the average SEP decays exactly as SNR to the power -m, identical to infinite-bit quantization.
  • When n exactly equals log2(M) the decay rate halves to SNR to the power -m/2.
  • When n is less than log2(M) the SEP approaches a positive constant independent of SNR.
  • For large m the addition of one extra bit above log2(M) produces a large reliability gain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • ADC bit-width decisions in fading links can be set directly from constellation size to preserve diversity order without full precision.
  • The result may guide resolution choices in other phase-modulated schemes that share similar decision-region geometry.
  • Extensions to non-circularly-symmetric fading or non-equiprobable symbols would test whether the same bit threshold remains sufficient.

Load-bearing premise

The wireless channel is subject to fading with a circularly-symmetric distribution and the signal points are equiprobable.

What would settle it

A plot of log(average SEP) versus log(SNR) whose slope equals m for n >= log2(M+1) but equals m/2 or zero for smaller n, under Nakagami-m fading.

Figures

Figures reproduced from arXiv: 1907.00789 by Hazer Inaltekin, Jamie Evans, Rajitha Senanayake, Samiru Gayan.

Figure 1
Figure 1. Figure 1: An illustration of the receiver architecture with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An illustration for the proof of Theorem 2, where [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average SEP as a function of SNR for 8-PSK and 16-PSK modulations. n = 2 < log2 M and m = 1. VI. CONCLUSIONS We have obtained fundamental performance limits, opti￾mum ML detectors and associated average SEP expressions for low-resolution ADC based communication systems. We have also performed an extensive numerical study to illustrate the accuracy of the derived analytical expressions. A ternary SEP behavi… view at source ↗
Figure 4
Figure 4. Figure 4: Average SEP curves as a function of SNR for different modulation schemes. n = log2 M, log2 M + 1, log2 M + 2 and m = 1. obtained by setting m = 1. The simulated results are again generated by using Monte Carlo simulations. We can clearly observe an error floor for high SNR values when n < log2 M in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: An illustration for the proof of Lemma 1 when [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

This paper considers a low-resolution wireless communication system in which transmitted signals are corrupted by fading and additive noise. First, a universal lower bound on the average symbol error probability (SEP), correct for all M-ary modulation schemes, is obtained when the number of quantization bits is not enough to resolve M signal points. Second, in the special case of M-ary phase shift keying (M-PSK), the optimum maximum likelihood detector for equiprobable signal points is derived. Third, utilizing the structure of the derived optimum receiver, a general average SEP expression for the M-PSK modulation with n-bit quantization is obtained when the wireless channel is subject to fading with a circularly-symmetric distribution. Finally, an extensive simulation study of the derived analytical results is presented for general Nakagami-m fading channels. It is observed that a transceiver architecture with n-bit quantization is asymptotically optimum in terms of communication reliability if n is greater than or equal to log_2(M +1). That is, the decay exponent for the average SEP is the same and equal to m with infinite-bit and n-bit quantizers for n greater than or equal to log_2(M+1). On the other hand, it is only equal to half and 0 for n = log_2(M) and n < log_2(M), respectively. Hence, for fading environments with a large value of m, using an extra quantization bit improves communication reliability significantly.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives a universal lower bound on average symbol error probability (SEP) for any M-ary modulation when the ADC resolution n is insufficient to resolve all signal points. For M-PSK it derives the optimum ML detector under equiprobable symbols and circularly-symmetric fading, obtains a general average SEP expression, and reports simulation results in Nakagami-m channels showing that the SEP decay exponent equals the fading parameter m (matching infinite-resolution ADCs) precisely when n ≥ log₂(M+1), equals ½ when n = log₂(M), and equals 0 when n < log₂(M).

Significance. If the diversity-order claims are placed on an analytical footing, the work would clarify the minimal quantization depth needed to preserve full diversity order in fading channels for PSK constellations, with direct implications for low-power transceiver design. The closed-form SEP expression under circularly-symmetric fading and the ML detector derivation constitute reusable technical contributions.

major comments (2)
  1. [Abstract / simulation results] Abstract and simulation section: the central claim that the SEP decay exponent equals m for n ≥ log₂(M+1) is stated as an observation from finite-SNR Monte-Carlo trials rather than from an explicit high-SNR asymptotic expansion of the derived SEP integral (e.g., leading-term behavior of the integrand near h=0). Without this expansion it remains possible that quantization boundaries alter the effective diversity order or introduce sub-dominant terms visible only at higher SNR.
  2. [SEP expression derivation] Section deriving the average SEP expression: the general integral form is obtained by averaging the conditional error probability over the circularly-symmetric fading density, yet the subsequent diversity-order statements rely on numerical fitting rather than an analytic argument that the quantization-induced decision regions preserve the same near-zero behavior of the integrand as the unquantized case when n ≥ log₂(M+1).
minor comments (2)
  1. [ML detector derivation] Notation: the precise definition of the n-bit quantizer thresholds and the mapping from received phase to decision regions should be stated explicitly before the ML detector derivation to avoid ambiguity in the subsequent SEP integral.
  2. [Simulation figures] Figure captions: several simulation plots compare quantized and unquantized SEP curves but do not indicate the exact SNR range used to extract the observed slopes; adding this information would strengthen reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. The points raised about strengthening the diversity-order claims with explicit asymptotics are valid, and we will revise the manuscript to address them directly.

read point-by-point responses

  1. Referee: [Abstract / simulation results] Abstract and simulation section: the central claim that the SEP decay exponent equals m for n ≥ log₂(M+1) is stated as an observation from finite-SNR Monte-Carlo trials rather than from an explicit high-SNR asymptotic expansion of the derived SEP integral (e.g., leading-term behavior of the integrand near h=0). Without this expansion it remains possible that quantization boundaries alter the effective diversity order or introduce sub-dominant terms visible only at higher SNR.

    Authors: We agree that the diversity-order claim would be stronger with an explicit high-SNR asymptotic analysis of the SEP integral rather than relying solely on simulations. In the revision we will derive the leading-term behavior of the integrand near h=0. Using the structure of the ML detector and the conditional error probability, we will show analytically that for n ≥ log₂(M+1) the quantization boundaries do not change the order of the singularity at h=0 relative to the unquantized case, thereby confirming the diversity order equals m. For the boundary cases n = log₂(M) and n < log₂(M) the leading behavior changes as observed in simulations. revision: yes

  2. Referee: [SEP expression derivation] Section deriving the average SEP expression: the general integral form is obtained by averaging the conditional error probability over the circularly-symmetric fading density, yet the subsequent diversity-order statements rely on numerical fitting rather than an analytic argument that the quantization-induced decision regions preserve the same near-zero behavior of the integrand as the unquantized case when n ≥ log₂(M+1).

    Authors: The referee is correct that the current diversity statements rest on numerical evidence. We will add an analytic argument in the revised manuscript. Starting from the derived integral expression for average SEP, we will examine the small-h expansion of the conditional SEP (which governs the high-SNR asymptotics). When n ≥ log₂(M+1) the ML decision regions ensure that the conditional error probability retains the same leading-order dependence on h as the infinite-resolution receiver; this directly implies the integrand near h=0 is unchanged and the diversity order remains m. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations self-contained from standard ML and fading integrals

full rationale

The paper first obtains a universal lower bound on average SEP for insufficient quantization bits, then derives the optimum ML detector for equiprobable M-PSK points under circularly-symmetric fading, and finally produces a general average SEP expression via the structure of that receiver. These steps follow directly from the definition of ML detection and integration against the fading pdf; none reduce to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation. The asymptotic exponent claim is explicitly labeled an observation from Nakagami-m simulations rather than an analytical high-SNR expansion, so it does not create a circular reduction. The provided text contains no self-citation chains or ansatzes imported from prior author work that would force the central result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities. The work implicitly relies on standard wireless assumptions such as additive noise and circularly-symmetric fading distributions, but these cannot be audited without the full manuscript.

pith-pipeline@v0.9.0 · 5800 in / 1239 out tokens · 31471 ms · 2026-05-25T13:52:23.513628+00:00 · methodology

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